Asymptotically Flat Structure of Hypergravity in Three Spacetime

CECS-PHY-15/03

Asymptotically ﬂat structure of hypergravity in three spacetime dimensions

Oscar Fuentealba,a,b Javier Matulich,a Ricardo Troncoso,a aCentro de Estudios Cientíﬁcos (CECs), Av. Arturo Prat 514, Valdivia, Chile. bDepartamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile.

E-mail: [email protected], [email protected], [email protected]

Abstract: The asymptotic structure of three-dimensional hypergravity without cosmological constant is analyzed. In the case of gravity minimally coupled to a spin-5/2 field, a consistent set of boundary conditions is proposed, being wide enough so as to accommodate a generic choice of chemical potentials associated to the global charges. The algebra of the canonical generators of the asymptotic symmetries is given by a hypersymmetric nonlinear extension of BMS3. It is shown that the asymptotic symmetry algebra can be recovered from a subset of a suitable limit of the direct sum of the W(2,4) algebra with its hypersymmetric extension. The presence of hypersymmetry generators allows to construct bounds for the energy, which turn out to be nonlinear and saturate for spacetimes that admit globally-defined “Killing vector-spinors”. The null orbifold or Minkowski spacetime can then be seen as the corresponding ground state in the case of fermions that fulfill periodic or antiperiodic boundary conditions, respectively. The hypergravity theory is also explicitly extended so as to admit parity-odd terms in the action. It is then shown that the asymptotic symmetry algebra includes an additional central charge, being proportional to the coupling of the Lorentz-Chern-Simons form. The generalization of these results in the case of gravity minimally coupled to arbitrary half-integer spin fields is also carried out. 1 The hypersymmetry bounds are found to be given by a suitable polynomial of degree s + 2 arXiv:1508.04663v2 [hep-th] 4 Nov 2015 in the energy, where s is the spin of the fermionic generators. Contents

1 Introduction1

2 General Relativity minimally coupled to a spin-5/2 ﬁeld2

3 Unbroken hypersymmetries: Killing vector-spinors4 3.1 Cosmological spacetimes and solutions with conical defects4

4 Asymptotically ﬂat behaviour and the hyper-BMS3 algebra6 4.1 Flat limit of the asymptotic symmetry algebra from the case of negative cosmological constant 10

5 Hypersymmetry bounds 12

6 Hypergravity reloaded 13

7 General Relativity minimally coupled to half-integer spin ﬁelds 15 7.1 Killing tensor-spinors 16 7.2 Asymptotically ﬂat structure and hypersymmetry bounds 17

8 Final remarks 20

A Conventions 24

B Killing vector-spinors from an alternative approach 24

1 C Hyper-Poincaré algebra with fermionic generators of spin n + 2 26

D Asymptotic hypersymmetry algebra 26 D.1 Spin-3/2 fields (supergravity) 26 D.2 Spin-7/2 fields 27 D.3 Spin-9/2 fields 29

1 Introduction

It has been shown that the inconsistencies arising in the minimal coupling of a massless spin-5/2 ﬁeld to General Relativity [1], [2], [3], [4] can be successfully surmounted in three-dimensional spacetimes [5]. This theory is known as hypergravity, and it has been recently reformulated as a Chern-Simons theory of a new extension of the Poincaré group with fermionic generators of spin 3/2 [6]. In the case of negative cosmological constant, additional spin-4 ﬁelds are required by consistency [7], [8], [9], and it can be seen that

– 1 – the anticommutator of the generators of the asymptotic hypersymmetries, associated to fermionic spin-3/2 parameters, leads to interesting nonlinear bounds for the bosonic global charges of spin 2 and 4 [9]. The bounds saturate provided the bosonic configurations admit globally-defined “Killing vector-spinors”. One of the main purposes of this paper is to show how these results extend to the case of asymptotically flat spacetimes in hypergravity, also in the case of arbitrary half-integer spin fields. In the next section we briefly sum- marize the formulation of hypergravity as a Chern-Simons theory for the hyper-Poincaré group in the simplest case of fermionic spin-5/2 fields, while section3 is devoted to explore the global hypersymmetry properties of cosmological spacetimes and solutions with conical defects. In the case of fermions that fulfill periodic boundary conditions, it is shown that the null orbifold possesses a single constant Killing vector-spinor. Analogously, for antiperiodic boundary conditions, Minkowski spacetime is singled out as the maximally (hyper)symmetric configuration, and the explicit expression of the globally-defined Killing vector-spinors is found. The asymptotically flat structure of hypergravity in three spacetime dimensions is analyzed in section4, where a precise set of boundary conditions that includes “chemical potentials” associated to the global charges is proposed. The algebra of the canonical generators of the asymptotic symmetries is found to be given by a suitable hypersymmetric nonlinear extension of the BMS3 algebra. It is also shown that this algebra corresponds to a subset of a suitable Inönü-Wigner contraction of the direct sum of the W(2,4) algebra with its hypersymmetric extension W 5 . The hypersymmetry bounds (2, 2 ,4) that arise from the anticommutator of the fermionic generators are found to be nonlinear, and are shown to saturate for spacetimes that admit unbroken hypersymmetries, like the ones aforementioned. This is explicitly carried out in section5. In section6, the previous analysis is performed in the case of an extension of the hypergravity theory that includes additional parity-odd terms in the action. It is found that the asymptotic symmetry algebra admits an additional central charge along the Virasoro subgroup. The results are then extended to the case of General Relativity minimally coupled to half-integer spin fields in section7, including the asymptotically flat structure, and the explicit expression of the Killing tensor-spinors. The hypersymmetry bounds are shown to be described by a polynomial of degree s + 1/2 in the energy, where s is the spin of the fermionic generators. We conclude in section8 with some final remarks, including the extension of these results to the case of hypergravity with additional parity-odd terms and fermions of arbitrary half-integer spin. The coupling of additional spin-4 fields is also addressed. AppendixA is devoted to our conventions, and in appendixB, an alternative interesting form to obtain the explicit form of the Killing vector-spinors is presented. The general form of the hyper-Poincaré algebra is discussed in appendixC, while appendixD includes the asymptotic hypersymmetry algebra in the case of fermionic fields of spin 3/2 (supergravity), as well as for fields of spin 7/2 and 9/2.

2 General Relativity minimally coupled to a spin-5/2 ﬁeld

It has been recently shown that the hypergravity theory of Aragone and Deser [5] can be reformulated as a gauge theory of a suitable extension of the Poincaré group with fermionic

– 2 – spin-3/2 generators [6]. The action is described by a Chern-Simons form, so that the dreibein, the (dualized) spin connection, and the spin-5/2 ﬁeld correspond to the components of a gauge ﬁeld given by

a a α a A = e Pa + ω Ja + ψa Qα , (2.1)

a that takes values in the hyper-Poincaré algebra, being spanned by the set {Pa,Ja,Qα}. a The fermionic ﬁelds and generators are assumed to be Γ-traceless, i. e., Γ ψa = 0, and a Q Γa = 0, so that the nonvanishing (anti)commutation rules read

c c [Ja,Jb] = εabcJ , [Ja,Pb] = εabcP ,

1 β c [Ja,Qαb] = (Γa) Qβb + εabcQ , (2.2) 2 α α n o 2 5 1 Qa ,Qb = − (CΓc) P ηab + εabcC P + (CΓ(a) P b) , α β 3 αβ c 6 αβ c 6 αβ where C stands for the charge conjugation matrix. The Majorana conjugate then reads ¯ β ψαa = ψa Cβα. Since the algebra admits an invariant bilinear form, whose only nonvanishing components are given by

D a b E 2 ab 1 abc hJa,Pbi = ηab , Q ,Q = Cαβη − ε (CΓc)αβ , (2.3) α β 3 3 the action can be written as k 2 I [A] = AdA + A3 , (2.4) 4π ˆ 3 which up to a surface term reduces to

k a ¯ a I = 2R ea + iψaDψ . (2.5) 4π ˆ

a a 1 abc Here R = dω + 2 ε ωbωc is the dual of the curvature two-form, and since the fermionic field is Γ-traceless, its Lorentz covariant derivative fulfills 1 Dψa = dψa + ωbΓ ψa + εabcω ψ 2 b b c a 3 b a a b = dψ + ω Γbψ − ωbΓ ψ . (2.6) 2 The field equations are then given by F = dA + A2 = 0, whose components read

a a 3 ¯ a b a R = 0 ,T = iψbΓ ψ , Dψ = 0 , (2.7) 4 where T a = Dea corresponds to the torsion two-form. Therefore, by construction, the action changes by a boundary term under local hy- α a persymmetry transformations spanned by δA = dA + [A, λ], with λ = a Qα, so that the transformation law of the ﬁelds reduces to

a 3 a b a a a δe = i¯bΓ ψ , δω = 0 , δψ = D . (2.8) 2 Note that the transformation rules of the ﬁelds in [5] agree with the ones in (2.8), on-shell.

– 3 – 3 Unbroken hypersymmetries: Killing vector-spinors

It is interesting to explore the set of bosonic solutions that possess unbroken global hypersymmetries. According to the transformation rules of the fields in (2.8), this class of configurations has to fulfill the following Killing vector-spinor equation:

a 1 b a abc d + ω Γb + ε ωbc = 0 , (3.1) 2 where the spin-3/2 parameter a is Γ-traceless. As it follows from the ﬁeld equations (2.7), the spin connection is locally ﬂat, and it a a −1 λ Ja can then be written as ω = ω Ja = g dg, with g = e . Therefore, the general solution of the Killing vector-spinor equation (3.1) is given by

α −1α b β a = gS β (gV )a ηb , (3.2)

β where ηb is a Γ-traceless constant vector-spinor. Here, gS and gV stand for the same group element g, but expressed in the spinor and the vector (adjoint) representations, respectively. Since the generators of the Lorentz group in the spinor and vector representations are given α 1 α by (Ja)β = 2 (Γa)β , and (Ja)bc = −εabc, they explicitly read α 1 a α a (gS) = exp λ (Γa) , (gV ) = exp [−λ εabc] . (3.3) β 2 β bc

Hence, bosonic conﬁgurations that admit unbroken hypersymmetries possess Killing vector-spinors of the form (3.2) provided they are globally well-deﬁned, either for periodic or antiperiodic boundary conditions.

3.1 Cosmological spacetimes and solutions with conical defects

Let us focus on an interesting class of circularly symmetric solutions that describe cosmological spacetimes as well as conﬁgurations with conical defects. The latter class was introduced in [10], [11] while the former one was explored in [12], [13], [14]. The thermo- dynamic properties of cosmological spacetimes have been analyzed in [15], [16], [17], [18]. As explained in [19], [20], [9], it is useful to express the solution for a ﬁxed range of the coordinates, so that the Hawking temperature and the chemical potential for the angular momentum manifestly appear in the metric. Hereafter we follow the conventions of [18], and for latter purposes, it is convenient to write the line element in outgoing null coordinates, which reads

2 2 2 4π πJ 2 2 2 2πµP J ds = − − P µ du − 2µP dudr + r dφ + µJ + du . (3.4) k kr2 P kr2

Here P determines the mass, whose associated “chemical potential” relates to the inverse −1 Hawking temperature according to µP = −β . Analogously, µJ stands for the chemical potential associated to the angular momentum J . We also assume a non-diagonal form for

– 4 – the Minkowski metric in a local frame, so that its nonvanishing components are given by η01 = η10 = η22 = 1. The dreibein can then be chosen as 2πµ P 2πJ e0 = −dr + P du + (dφ + µ du) , e1 = µ du , e2 = r (dφ + µ du) , k k J P J (3.5) and hence, the components of the dualized spin connection are given by

0 2πP 1 2 ω = (dφ + µJ du) , ω = dφ + µJ du , ω = 0 . (3.6) k As explained at the beginning of section3, since the curvature two-form vanishes, the spin connection (3.6) is locally ﬂat, and it can then be generically written as ω = g−1dg, with 2πP ˆ g = exp J1 + J0 φ , (3.7) k

ˆ and φ = φ + µJ u. Note that in the case of P 6= 0, for the spinor and vector representations, the group element g in (3.7) exponentiates as

"r # r "r # πP ˆ k πP ˆ 2πP gS = cosh φ 2×2 + sinh φ J1 + J0 , (3.8) k I πP k k

r " r # "r #2 2 1 k πP 2πP k πP 2πP g = + sinh 2 φˆ J + J + sinh φˆ J + J , V I3×3 2 πP k 1 k 0 2πP k 1 k 0 (3.9) respectively, while for P = 0, it reduces to

ˆ ˆ 1 ˆ2 2 gS = 2×2 + φJ1 , gV = 3×3 + φJ1 + φ J . (3.10) I I 2 1 One then concludes that cosmological spacetimes, for which P > 0, necessarily break all the hypersymmetries. Indeed, this class of solutions cannot admit globally-defined Killing vector-spinors because, according to (3.8) and (3.9), the (anti)periodic boundary conditions for the vector-spinor a in (3.2) fail to be fulfilled. In the case of configurations with P = 0, equations (3.2) and (3.10) imply that the Killing vector-spinor is constant and satisfies: 3 Γ1a − Γa1 = 0 , (3.11) 2 so that it fulfills periodic boundary conditions, and possesses a single nonvanishing compo- − − nent given by 0 = η0 . For the remaining case, P := −kj2/π < 0, describing solutions with conical defects, the group element in both representations reduces to

h ˆi 1 h ˆi 2 gS = cos jφ 2×2 + sin jφ J1 − 2j J0 , (3.12) I j

– 5 – 2 1 h ˆi 2 1 h ˆi 2 2 gV = 3×3 + sin 2jφ J1 − 2j J0 + sin jφ J1 − 2j J0 . (3.13) I 2j 2j2

Therefore, this class of configurations possesses four independent Killing vector-spinors that fulfill (anti)periodic boundary conditions provided j is a (half-)integer. The explicit form of the Killing vector-spinors is then obtained from (3.2), where gS and gV are given by eqs. (3.12) and (3.13). Note that this is the maximum number of hypersymmetries. Indeed, for these configurations the holonomy of the spin connection becomes trivial, which −1 ˆ ˆ in the spinor representation means that gS (φ)gS(φ + 2π) = −I2×2, while in the vector −1 ˆ ˆ representation the condition reads gV (φ)gV (φ + 2π) = I3×3. It is worth pointing out that if j were different from a (half-)integer, the configurations would not solve the field equations in vacuum. This is because they would possess a conical singularity at the origin, and hence they should necessarily be supported by an external source. As it occurs in the case of supersymmetry, it is natural to expect that the bosonic global charges fulfill suitable bounds that turn out to be saturated for configurations that possess unbroken hypersymmetries. Indeed, as shown in [21], the bounds that correspond to three-dimensional supergravity with asymptotically flat boundary conditions certainly do so. Actually, the bounds also exclude conical surplus solutions, in particular those whose angular coordinate ranges from zero to 4πj, with j > 1/2, despite they are maximally supersymmetric. When a negative cosmological constant is considered, this is also the case not only for supergravity [22], but also for hypergravity [9], where in the latter case the bounds turn out to be nonlinear. Thus, one of the main purposes of the following sections is showing how these results can be extended to the case of hypergravity endowed with a suitable set of asymptotically flat boundary conditions, as well as how to recover them in the vanishing cosmological constant limit.

4 Asymptotically ﬂat behaviour and the hyper-BMS3 algebra

Let us introduce a suitable set of asymptotic conditions that allows to describe the dynamics of asymptotically flat spacetimes in hypergravity. The set must be relaxed enough so as to accommodate the solutions of interest that have been described in section 3.1, and simultaneously, restricted in an appropriate way in order to ensure finiteness of the canonical generators associated to the asymptotic symmetries. In the case of pure General Relativity, a consistent set of boundary conditions indeed exists, whose asymptotic symmetry algebra corresponds to BMS3 with a nontrivial central extension [23], [24], [25]. These results have been extended to the case of supergravity [21], as well as for General Relativity coupled to higher spin fields [26], [27], [17], [18]. In order to carry out this task in hypergravity, we take advantage of the Chern-Simons formulation of the theory, depicted in section2. Since the hypersymmetry generators are Γ−traceless, it is useful to get rid of Q2 = Q1Γ0 − Q0Γ1,

– 6 – so that once the remaining generators are relabeled according to

Jˆ−1 = −2J0 , Jˆ1 = J1 , Jˆ0 = J2 ,

Pˆ−1 = −2P0 , Pˆ1 = P1 , Pˆ0 = P2 , 5 √ 3 √ ˆ 4 ˆ 4 Q− 3 = 2 3Q+0 , Q− 1 = 2 3Q−0 , (4.1) 2 √ 2 √ 1 − 1 Qˆ 1 = −2 4 3Q+1 , Qˆ 3 = −2 4 3Q−1 , 2 2 the hyper-Poincaré algebra (2.2) reads

h i Jˆm, Jˆn = (m − n) Jˆm+n , h i Jˆm, Pˆn = (m − n) Pˆm+n , h i 3m Jˆm, Qˆp = − p Qˆm+p , (4.2) 2 n o 1 Qˆ , Qˆ = 6p2 − 8pq + 6q2 − 9 Pˆ , p q 4 p+q

1 3 with m, n = ±1, 0, and p, q = ± 2 , ± 2 . Thus, following the lines of [28], and as explained in [21], [27], the radial dependence of the asymptotic form of the gauge ﬁeld can be gauged away by a suitable group element r Pˆ of the form h = e 2 −1 , so that

A = h−1ah + h−1dh , (4.3) and hence, the remaining analysis can be entirely performed in terms of the connection a = audu + aφdφ, that depends only on time and the angular coordinate. As explained in [19], [20], one starts prescribing the asymptotic form of the dynamical gauge field at a fixed time slice with u = u0, so that the asymptotic fall-off of aφ is assumed to be such that the deviations with respect to the reference background go along the highest weight generators of (4.2). Choosing the reference background to be given by the null orbifold [29], that corresponds to the configuration in (3.4) with J = P = 0, the asymptotic form of the dynamical field reads

ˆ π ˆ ˆ ψ ˆ aφ = J1 − J P−1 + PJ−1 − Q− 3 , (4.4) k 3 2 where J , P and ψ stand for arbitrary functions of u, φ. The asymptotic symmetries then correspond to gauge transformations δa = dλ + [a, λ] that preserve the form of (4.4). Therefore, the hyper-Poincaré-valued parameter λ is found to depend on three arbitrary functions of u and φ, so that

ˆ ˆ ˆ λ = T P1 + Y J1 + EQ 3 + η 3 [T,Y, E] , (4.5) 2 ( 2 )

– 7 – where E is Grassmann-valued, and 0 ˆ 0 ˆ 0 ˆ 1 2π 00 ˆ η 3 [T,Y, E] = −T P0 − Y J0 − E Q 1 − Y P − Y J−1 ( 2 ) 2 2 k π 3 k 00 ˆ 1 3π 00 ˆ − T P + Y J − iψE − T P−1 − EP − E Q− 1 k 2 2π 2 k 2 π 7 0 3 0 k 000 ˆ − Y ψ − E P − EP + E Q− 3 ; (4.6) 3k 2 2 2π 2 while the transformation law of the ﬁelds reads k δP = 2PY 0 + P0Y − Y 000 , 2π k 5 3 δJ = 2J Y 0 + J 0Y + 2PT 0 + P0T − T 000 + iψE0 + iψ0E , (4.7) 2π 2 2 5 9π 3 k δψ = ψY 0 + ψ0Y − P2E + P00E + 5P0E0 + 5PE00 − E0000 . 2 2k 2 2π

Hereafter, prime stands for ∂φ. Since the time evolution of aφ corresponds to a gauge transformation parametrized by the Lagrange multiplier au, its asymptotic form will be maintained along diﬀerent time slices provided au is of the allowed form, i. e.,

au = λ [µP , µJ , µψ] , (4.8) where the chemical potentials µP , µJ , µψ stand for arbitrary functions of u, φ, that are assumed to be fixed at the boundary. Consistency then demands that the field equations, which now reduce to k P˙ = 2Pµ 0 + P0µ − µ 000 , J J 2π J ˙ 0 0 0 0 k 000 5 0 3 0 J = 2J µJ + J µJ + 2PµP + P µP − µP + iψµψ + iψ µψ , (4.9) 2π 2 2 5 9π 3 k ψ˙ = ψµ 0 + ψ0µ − P2µ + P00µ + 5P0µ 0 + 5Pµ 00 − µ 0000 , 2 J J 2k ψ 2 ψ ψ ψ 2π ψ have to hold in the asymptotic region, while the parameters of the asymptotic symmetries fulfill the following conditions

0 0 Y˙ = µJ Y − µJ Y, ˙ 0 0 0 0 9π 3 00 0 0 3 00 T = µJ T − µJ T + µP Y − µP Y + iµψEP − iµψ E + 2iµψ E − iµψE ,(4.10) k 2 2 3 3 E˙ = µ Y 0 − µ 0Y − µ 0E + µ E0 , 2 ψ ψ 2 J J which are needed in order to ensure that the global charges are conserved.1 Following the Regge-Teitelboim approach [30], the variation of the canonical generators is found to be generically given by k δQ [λ] = − hλδaφi dφ , (4.11) 2π ˆ

1Since global symmetries are necessarily contained within the asymptotic ones, these results provide an interesting alternative path to ﬁnd the explicit expression of the Killing vector-spinors. See appendixB.

– 8 – which by virtue of (4.4) and (4.5), up to an arbitrary constant without variation, integrate as Q [T,Y, E] = − (T P + Y J − iEψ) dφ . (4.12) ˆ It is worth highlighting that the global charges are manifestly independent of the radial coordinate r. Therefore, the boundary can be located at an arbitrary ﬁxed value r = r0, and it corresponds to a timelike surface with the topology of a cylinder.

Since the Poisson brackets fulﬁll {Q [λ1] ,Q [λ2]} = δλ2 Q [λ1], the algebra of the canonical generators can be directly obtained from the transformation law of the ﬁelds in (4.7). 1 P inφ Expanding in Fourier modes, X = 2π n Xne , the nonvanishing Poisson brackets read

i {Jm, Jn} = (m − n) Jm+n , 3 i {Jm, Pn} = (m − n) Pm+n + km δm+n,0 , 3m i {Jm, ψn} = − n ψm+n , (4.13) 2 1 9 X i {ψ , ψ } = 3m2 − 4mn + 3n2 P + P P + km4δ , m n 2 m+n 4k m+n−q q m+n,0 q where the modes of the generators ψm are labeled by (half-)integers when the fermions fulﬁll (anti)periodic boundary conditions. It is then clear that, with respect to Jm, the conformal weight of the generators Pm and ψn, is given by 2 and 5/2, respectively. Note that the subset spanned by Jm and Pm corresponds to the BMS3 algebra of General Relativity with the same central extension, and hence (4.13) stands for its hypersymmetric extension that is manifestly nonlinear. It is useful to perform the following shift in the generators: k Pn → Pn − δn,0 , (4.14) 2 so that the algebra now reads

i {Jm, Jn} = (m − n) Jm+n , 2 i {Jm, Pn} = (m − n) Pm+n + km m − 1 δm+n,0 , 3m i {Jm, ψn} = − n ψm+n , (4.15) 2 1 9 X i {ψ , ψ } = 6m2 − 8mn + 6n2 − 9 P + P P m n 4 m+n 4k m+n−q q q 9 1 +k m2 − m2 − δ , 4 4 m+n,0 in agreement with the result that has been recently anticipated in [6]. Indeed, dropping the nonlinear terms in (4.15), when the fermions fulﬁll antiperiodic boundary conditions, the wedge algebra, which is spanned by the subset of {Jm, Pm, ψn} with m = ±1, 0 and n = ±3/2, ±1/2, reduces to the hyper-Poincaré algebra in eq. (4.2).

– 9 – It can also be seen that the hyper-BMS3 algebra (4.13) turns out to be a subset of a precise Inönü-Wigner contraction of the direct sum of the W(2,4) algebra with its hypersymmetric extension W 5 . This is the main subject of the next subsection. (2, 2 ,4)

4.1 Flat limit of the asymptotic symmetry algebra from the case of negative cosmological constant

It has been recently shown that the asymptotic symmetries of three-dimensional hypergravity with negative cosmological constant are spanned by two copies of the classical limit of the WB2 algebra [9]. This algebra is also known as W 5 and corresponds to the hyper- (2, 2 ,4) symmetric extension of W(2,4) [31], [32]. The hypergravity theory that was discussed in [9] possesses the minimum number of hypersymmetries in each sector, so that the gauge group is given by OSp (1|4)⊗OSp (1|4). In analogy with the case of three-dimensional supergravity [33], one may say that the theory aforementioned corresponds to the N = (1, 1) AdS3 hypergravity. In this sense, there are two inequivalent minimal locally hypersymmetric extensions of General Relativity with negative cosmological constant, which correspond to the (1, 0) and the (0, 1) theories. It is then simple to verify that both minimal theories possess the same vanishing cosmological constant limit, and hence in order to proceed with the analysis we will consider the (0, 1) one, whose gauge group is given by Sp (4) ⊗ OSp (1|4). According to [9], the asymptotic symmetry algebra of the minimal hypergravity theory with negative cosmological constant then corresponds to W(2,4)⊕W 5 . (2, 2 ,4) The classical limit of the W 5 algebra reads (2, 2 ,4)

κ i {L , L } = (m − n) L + m3δ , m n m+n 2 m+n,0 i {Lm, Un} = (3m − n) Um+n , 3m i {L , Ψ } = − n Ψ , m n 2 m+n 1 i {U , U } = (m − n) 3m4 − 2m3n + 4m2n2 − 2mn3 + 3n4 L m n 2232 m+n 3 1 2 2 2 3π (6) + (m − n) m − mn + n Um+n − (m − n)Λ (4.16) 6 κ m+n 72π κ − (m − n) m2 + 4mn + n2 Λ(4) + m7δ , 32κ m+n 2332 m+n,0 1 23π i {U , Ψ } = m3 − 4m2n + 10mn2 − 20n3 Ψ − iΛ(11/2) m n 223 m+n 3κ m+n π + (23m − 82n)Λ(9/2) , 3κ m+n 1 4 3π κ i {Ψ , Ψ } = U + m2 − mn + n2 L + Λ(4) + m4δ , m n m+n 2 3 m+n κ m+n 6 m+n,0 where the fermionic modes are labeled by (half-)integers in the case of (anti)periodic bound- (l) (l) −imφ ary conditions, and Λm = Λ e dφ stand for the mode expansion of the nonlinear ´

– 10 – terms, given by Λ(4) = L2 , (4.17) Λ(9/2) = LΨ , (4.18) 27 Λ(11/2) = L0Ψ , (4.19) 23 7 8π 295 2 22 25 Λ(6) = − UL − L3 + (L0) + L00L + iΨΨ0 . (4.20) 18 3κ 432 27 12

The bosonic generators Lm and Um span the W(2,4) subalgebra. In order to take the vanishing cosmological constant limit of the asymptotic symmetry algebra of the minimal theory, given by W(2,4)⊕W 5 , it is useful to perform the following (2, 2 ,4) change of basis: 1 P = L+ + L− , J = L+ − L− , n ` n −n n n −n r 1 + − 1 + − 6 + Wn = √ U + U , Vn = √ U − U , ψn = Ψ , (4.21) ` n −n ` n −n ` n − − + + + where Ln , Un stand for the generators of the (left) W(2,4) algebra, and Ln , Un , ψn span the (right) W 5 algebra. Therefore, rescaling the level according to κ = k`, in the large (2, 2 ,4) AdS radius limit, ` → ∞, one obtains that the nonvanishing brackets of the contracted algebra read

i {Jm, Jn} = (m − n) Jm+n , 3 i {Jm, Pn} = (m − n) Pm+n + km δm+n,0 ,

i {Jm, Wn} = (3m − n) Wm+n ,

i {Jm, Vn} = (3m − n) Vm+n , 1 4 3 2 2 3 4 i {Vm, Wn} = (m − n) 3m − 2m n + 4m n − 2mn + 3n Pm+n (4.22) 2232 3 2 2 π (6) 7 X − (m − n) Λ˜ − (m − n) m2 + 4mn + n2 P P k m+n 324k m+n−q q q k + m7δ , 2232 m+n,0 3m i {J , ψ } = − n ψ , m n 2 m+n 1 9 X i {ψ , ψ } = 3m2 − 4mn + 3n2 P + P P + km4δ , m n 2 m+n 4k m+n−q q m+n,0 q with 7 2π 295 2 11 Λ˜ (6) = − WP − P3 + (P0) + PP00 . (4.23) 12 k 288 9 It is then apparent that one can consistently get rid of the (conformal) spin-4 generators Vm, Wn, since the Inönü-Wigner contraction of W(2,4)⊕W 5 in eq. (4.22) possesses a (2, 2 ,4) subset spanned by {Pm, Jm, ψm}, which precisely corresponds to the hyper-BMS3 algebra in (4.13). Note that this is just a reﬂection of the fact that in the vanishing cosmological constant limit, the hypergravity theory can be consistently formulated without the need of spin-4 ﬁelds.

– 11 – 5 Hypersymmetry bounds

In the case of hypergravity with negative cosmological constant, it has been recently shown that the anticommutator of the generators of the asymptotic hypersymmetries implies the existence of interesting nonlinear bounds for the bosonic charges, that saturate for configurations that admit unbroken hypersymmetries [9]. In this section, following these lines, we explicitly show that this is also the case for hypergravity with asymptotically flat boundary conditions. In order to perform this task, it is useful to assume that the bosonic global charges are just determined by the zero modes. Indeed, as explained in [20], a generic bosonic configuration can be brought to the “rest frame” through the action of suitable elements of the asymptotic symmetry algebra. The searched for bounds can then be found along the same semi-classical reasoning as in the case of supergravity [34], [35], [36], [37], [38], [39], [22]. Hence, the fermionic bracket in (4.13) becomes an anticommutator, which in the rest frame, and for m = −n = p, reads

1 ˆ ˆ ˆ ˆ 2 9π 2 k 4 ψpψ−p + ψ−pψp = 5p Pˆ + Pˆ + p ≥ 0 , (5.1) 2π 2k 2π with Pˆ0 = 2πPˆ. Thus, since the left-hand side of (5.1) is a positive-definite hermitian operator, in the classical limit, and for any value of the (half-)integer p, the energy has to fulfill the following bounds: 9π π p2 + P p2 + P ≥ 0 , (5.2) k k which are manifestly nonlinear. Note that for any configuration with P > 0, the bounds in (5.2) are automatically fulfilled, but never saturate. Indeed, this is the case of the cosmological spacetimes in (3.4), which goes by hand with the fact that they do not admit globally-defined Killing vector-spinors, and hence, break all the hypersymmetries. These bounds are also clearly fulfilled in the case of P = 0, and for fermions with periodic boundary conditions, the one for p = 0 is saturated. This relates to the fact that this class of configurations, that includes the null orbifold, possesses a single unbroken hypersymmetry spanned by a constant Killing vector-spinor. In the case of P < 0, the class of smooth configurations are the ones for which the holonomy of the connection around an angular cycle is trivial. This means that they are maximally (hyper)symmetric, and then possess four Killing vector-spinors. As explained in section 3.1, their energy is given by P = −kj2/π, and the bounds in (5.2) then reduce to

2 2 2 2 p − 9j p − j ≥ 0 . (5.3)

Remarkably, the bounds are only fulﬁlled in the case of j2 = 1/4, so that four of them saturate, corresponding to p = ±1/2, and p = ±3/2. This is the case of Minkowski spacetime (P = −k/4π), with four independent Killing vector-spinors that fulﬁll antiperiodic boundary conditions. Hence, in spite of being maximally hypersymmetric, smooth solutions whose energy is lower than the one of Minkowski spacetime are excluded by the hypersymmetry bounds.

– 12 – It is worth highlighting that one arrives to similar conclusions in the case of asymptotically flat spacetimes in supergravity [21]. In fact, despite the analysis is fairly different, the supersymmetry bounds precisely select the same spectrum, including the corresponding ground states who saturate the bounds for spinors that fulfill different periodicity conditions.

6 Hypergravity reloaded

Let us look for a different theory of three-dimensional (hyper)gravity that is still compatible with the asymptotically flat boundary conditions described above, but now allowing the presence of spacetime torsion even in vacuum. For simplicity, we consider modifications such that the field equations are still of first order for the dreibein and the spin connection. One interesting possibility is to include additional terms, so that the action is given by

k a 2 a b c a Iγ = 2R ea + γ abce e e + 2γT ea . (6.1) 4π ˆ

Remarkably, despite the second (volume) term in the action looks like a cosmological constant Λ = −3γ2, the ﬁeld equations actually imply that the Riemann curvature vanishes. Indeed, the presence of the last (parity-odd) term in the action has the eﬀect of making the volume term to act as the source for a fully antisymmetric torsion in vacuum, being proportional to the volume element, given by

a abc T = −γε ebec , (6.2) so that the remaining ﬁeld equations ﬁx the curvature two-form according to

a 1 2 abc R = γ ε ebec . (6.3) 2 Therefore, eq. (6.2) implies that the spin connection splits as ωa =ω ¯a +κa, where ω¯a is the (torsionless) Levi-Civita connection, and the contorsion reads κa = −γea. The curvature a ¯a 1 2 abc two-form is then given by R = R + 2 γ ε ebec, and hence, equation (6.3) implies the ¯ 1 ρτ µ ν vanishing of the Riemann tensor, i. e., Ra = 4 εaρτ R µνdx dx = 0. The most general theory that possesses the features described above is obtained by a 1 a b c considering the addition of the Lorentz-Chern-Simons form, L(ω) = ω dωa + 3 εabcω ω ω , with an independent coupling µ, provided the remaining couplings in (6.1) are suitable shifted. The action is then given by

k a 2 γ a b c a Iµ,γ = 2 (1 + µγ) R ea + γ 1 + µ abce e e + µL(ω) + γ (2 + µγ) T ea . (6.4) 4π ˆ 3

Noteworthy, despite the fact that the Lorentz-Chern-Simons form is not a boundary term, the shifts in the other couplings are such that the field equations in vacuum just become reshuffled, coinciding with the previous ones for µ = 0, given by (6.2) and (6.3). Actually, one should highlight that both actions, (6.1) and (6.4), differ off-shell, which reflects through the fact that the canonical generators do not have the same form. Consequently, as in

– 13 – the case of supergravity [21], the asymptotic symmetry algebra of the latter acquires an additional central extension with respect to the former (see below). The locally hypersymmetric extension of the theory described by (6.4) is given by the following action k ¯ 3 b a Iµ,γ,ψ = Iµ,γ + iψa D + γe Γb ψ , (6.5) 4π ˆ 2 which is invariant under the following local hypersymmetry transformations:

a 3 a b a 3 a b a a 3 b a a b δe = i¯bΓ ψ , δω = − iγ¯bΓ ψ , δψ = D + γe Γb − γebΓ . (6.6) 2 2 2 The ﬁeld equations now read 1 3 3 Ra = γ2εabce e − iγψ¯ Γaψb ,T a = −γεabce e + iψ¯ Γaψb, 2 b c 4 b b c 4 b

a 3 b a a b Dψ = − γe Γbψ + γebΓ ψ . (6.7) 2 Note that in the case of µ = γ = 0, the action (6.5), the transformations rules (6.6), and the field equations (6.7), reduce to the ones of the locally hypersymmetric extension of General Relativity, given by eqs. (2.5), (2.8), and (2.7), respectively. As outlined in [6], in analogy with the case of supergravity [40], [21], the action (6.4) can be formulated as a Chern-Simons one for the hyper-Poincaré group (2.2) by virtue of a simple modification of the invariant bilinear form, and a suitable shift of the spin connection. Indeed, the invariant bilinear form in (2.3) can be consistently modified to admit an additional nonvanishing component given by

hJa,Jbi = µηab , (6.8) so that the Chern-Simons action (2.4) now depends on a different hyper-Poincaré-algebra- valued gauge field, defined as

a a α a A = e Pa +ω ˆ Ja + ψa Qα , (6.9) with ωâ := ωa + γea. Therefore, in terms of the covariant derivative with respect to ωâ and its corresponding curvature, given by Dˆ and Râ, respectively, up to a surface term, the Chern-Simons action reduces to

k â ¯ ˆ a Iµ,γ,ψ = 2R ea + µL(ˆω) + iψaDψ , (6.10) 4π ˆ which precisely agrees with (6.5). Note that the field equations (6.7) correspond to the vanishing of the components of the curvature associated to (6.9), so that they can be compactly written as F = dA + A2 = 0, being manifestly covariant under the full hyper- Poincaré group. One of the advantages of having formulated the extension of hypergravity with parity- odd terms as a Chern-Simons theory, is that its asymptotically flat structure can be directly obtained along the lines of the results in section4.

– 14 – The asymptotically flat boundary conditions for the connection (6.9) are then proposed to be precisely as in eqs. (4.3), (4.4), and (4.8), so that the asymptotic fall-off of the spin connection ωa becomes modified. Therefore, the asymptotic symmetries remain the same as in section4, being spanned by the hyper-Poincaré algebra valued parameter λ = λ [T,Y, E] given by (4.5). The global charges are then found to acquire a correction due to the additional component of the invariant bilinear form in (6.8), so that they now read Q [T,Y, E] = − T P + Y J˜ − iEψ dφ , (6.11) ˆ with J˜=J +µP, and do not depend on the parameter γ. Note that the shift in the canonical generator associated to Y implies that in the extended theory, even static configurations, as it is the case of Minkowski spacetime, may carry angular momentum. It is then simple to verify that, once the canonical generators are expanded in modes, their nonvanishing Poisson brackets are given by

n o 3 i J˜m, J˜n = (m − n) J˜m+n + µkm δm+n,0 ,

n o 3 i J˜m, Pn = (m − n) Pm+n + km δm+n,0 , n o 3m i J˜m, ψn = − n ψm+n , (6.12) 2 1 9 X i {ψ , ψ } = 3m2 − 4mn + 3n2 P + P P + km4δ , m n 2 m+n 4k m+n−q q m+n,0 q which corresponds to a hypersymmetric extension of the BMS3 algebra, with an additional central extension along its Virasoro subalgebra.

7 General Relativity minimally coupled to half-integer spin ﬁelds

3 In the generic case of fermionic ﬁelds of spin n + 2 , and in the absence of cosmological constant, the hypergravity action reads [5], [6] k a ¯ a1...an I = 2R ea + iψa ...a Dψ , (7.1) 4π ˆ 1 n

a1 where ψa1...an describes a Grassmann-valued 1-form that is Γ−traceless, i. e., Γ ψa1...an = 0, and completely symmetric in its vector indices. Its covariant derivative can be conveniently written as 1 a1...an a1...an b a1...an (a1 a2...an)b Dψ = dψ + n + ω Γbψ − ωbΓ ψ . (7.2) 2

The standard supergravity action in [41], [42], [43] is then recovered for n = 0, while the theory discussed in section2 corresponds to n = 1. The generic theory can also be formulated in terms of a Chern-Simons action for a gauge ﬁeld that takes values in the hyper-Poincaré algebra, given by

a a α a1...an A = e Pa + ω Ja + ψa1...an Qα . (7.3)

– 15 – a1...an 1 Here Qα correspond to Γ−traceless fermionic generators of spin n + 2 . The explicit expression of the generic hyper-Poincaré algebra can be compactly written in terms of its Maurer-Cartan form (see appendixC). The ﬁeld equations then read F = dA + A2 = 0, with a ˜a α a1...an F = R Ja + T Pa + Dψa1...an Qα , (7.4) where the hypercovariant torsion is now given by 1 1 a a ¯ a a1...an T˜ = T − n + iψa ...a Γ ψ . (7.5) 2 2 1 n

Thus, by construction, the action is invariant under gauge transformations generated by α a1...an λ = a1...an Qα , so that 1 δea = n + i¯ Γaψa1...an , 2 a1...an δωa = 0 , (7.6) δψa1...an = Da1...an .

7.1 Killing tensor-spinors According to (7.6), a purely bosonic conﬁguration is invariant under local hypersymmetry transformations provided the following “Killing tensor-spinor equation” is fulﬁlled: 1 b b da ...a + n + ω Γba ...a − ω Γ = 0 . (7.7) 1 n 2 1 n (a1 a2...an)b

Since the field equations imply the vanishing of the curvature two-form Ra, the general solution of (7.7) is now given by α α = g−1 (g )b1 ··· (g )bn ηβ , a1···an S β V a1 V an b1···bn (7.8) where gS and gV are defined in eq. (3.3). As explained in section3, both stand for the same group element g that determines the spin connection, ω = g−1dg, but expressed in ηβ the spinor and the vector representations, respectively. In the generic case, b1···bn is a constant Γ−traceless tensor-spinor. Unbroken hypersymmetries then correspond to Killing tensor-spinors of the form (7.8), that are globally well-defined. The hypersymmetry properties of the class of solutions discussed in section 3.1, describing cosmological spacetimes and configurations with conical defects, then go as follows. For any configuration with P= 6 0, gS and gV are given by (3.8) and (3.9), respectively; while in the case of P = 0, they read as in eq. (3.10). Therefore, in the case of P > 0 the solutions α cannot possess globally-defined Killing tensor-spinors, because a1···an in (7.8) do not fulfill neither periodic nor antiperiodic boundary conditions. This means that hypersymmetries are necessarily broken for cosmological spacetimes. By virtue of (7.8) and (3.10), configurations with P = 0 only admit constant Killing tensor-spinors that fulfill the following condition: 1 n + Γ1a ...a − Γ = 0 , (7.9) 2 1 n (a1 a2...an)1

– 16 – − − which implies that they have a single nonvanishing component given by 00···0 = η00···0. Therefore, this class of spacetimes possesses just one unbroken hypersymmetry, which relates to the fact that there is only one hypersymmetry bound that saturates for fermions with periodic boundary conditions (see below). As explained in section 3.1, smooth solutions with conical defects are maximally hypersymmetric and their energy is determined by P = −kj2/π < 0, where j is a (half-)integer. For this class of conﬁgurations, the explicit form of the Killing tensor-spinors is then given by (7.8), with gS and gV being described by eqs. (3.12) and (3.13), respectively. It will also be shown below that conical surpluses are excluded by the hypersymmetry bounds, which are fulﬁlled only for j2 = 1/4, which corresponds to the case of Minkowski spacetime.

7.2 Asymptotically flat structure and hypersymmetry bounds In order to describe the asymptotically flat behaviour of hypergravity in the generic case, it is convenient to make use of the Γ−traceless condition of the fields and the generators, which amounts to reduce the number of independent components. The hyper-Poincaré algebra in (C.2) can then be alternatively written as h i Jˆm, Jˆn = (m − n) Jˆm+n h i Jˆm, Pˆn = (m − n) Pˆm+n , h i Jˆm, Qˆp = (sm − p) Qˆm+p , (7.10) n ˆ ˆ o (s) ˆ Qp, Qq = fp,q Pp+q ,

1 3 with m, n = 0, ±1, and p, q = ± 2 , ± 2 ..., ±s, where s stands for the spin of the fermionic ˆ (s) (s) (s) generators Qp. The structure constants fulﬁll fp,q = fq,p = f−p,−q, and the nonvanishing ones are given by |p| (s) 2p (s) p+ 1 Y (2s + 2k) f = − f = (−1) 2 2p , (7.11) p,−p s + p + 1 p,−p−1 (2s − 2 (k − 1)) 1 k= 2 provided |p + q| ≤ 1. Here the fermionic generators have been normalized according to (s) f 1 1 = −1. 2 ,− 2 It is amusing to verify that the Jacobi identity now translates into the fact that the (s) structure constants fp,q solve the following recursion relation: (s) (s) (s) (m − (p + q)) fq,p − (sm − p) fq,m+p − (sm − q) fp,m+q = 0 . (7.12) For later purposes it is useful to note that

(s) s+ 1 2s (4s)!! f = (−1) 2 . (7.13) s,−s 2s + 1 (2s − 1)!!2 The structure constants can also be conveniently written as

1 s− 2 (s) X (l) fm,n = hm,n , (7.14) l=0

– 17 – (l) (l) 2l (l) where hm,n stand for homogeneous polynomials of degree 2l in m, n, i. e., hλm,λn = λ hm,n. Indeed, as it is shown below, the asymptotic symmetry algebra can be naturally expressed (l) in terms of hm,n, where m, n are extended to be arbitrary (half-)integers. Note that in the (1/2) case of supergravity fm,n = −1, while for fermionic generators of spin s = 3/2, the form (3/2) of fm,n can be read from eq. (4.2). In the case of fermionic generators with s = 5/2, 7/2 (5/2) (7/2) the explicit form of fm,n and fm,n is given in appendices D.2 and D.3, respectively. Following the lines of section4, the asymptotic form of the gauge field can be written as in eq. (4.3), so that at a fixed time slice, the dynamical field is proposed to be given by ˆ π ˆ ˆ ˆ aφ = J1 − J P−1 + PJ−1 + αsψQ−s , (7.15) k with (s) −1 (s) −1 αs = f−s,s+1 = −2s fs,−s , (7.16) (s) and fs,−s can be read from eq. (7.13). The asymptotic symmetries are then generically spanned by a hyper-Poincaré-valued parameter of the form ˆ ˆ ˆ λ = T P1 + Y J1 + EQs + η(s) [T,Y, E] , (7.17) where η(s) [T,Y, E] goes along all but the lowest weight generators, provided the fields J , P, ψ transform in a suitable way. The asymptotic form of the Lagrange multiplier can then be written in terms of the chemical potentials according to

au = λ [µP , µJ , µψ] . (7.18)

Its form is preserved under evolution in time as long as the field equations are fulfilled in the asymptotic region, and the parameters are subject to appropriate conditions, being described by first order equations in time. In order to integrate the variation of the canonical generators in (4.11), one needs the relevant fermionic component of the invariant bilinear form, which is given by D ˆ ˆ E −1 Qs, Q−s = 2αs , (7.19) so that the global charges in the generic case acquire the same form as in eq. (4.12), i. e.,

Q [T,Y, E] = − (T P + Y J − iEψ) dφ . (7.20) ˆ Once expanded in Fourier modes, the nonvanishing Poisson brackets of the canonical generators are given by i {Jm, Jn} = (m − n) Jm+n , 3 i {Jm, Pn} = (m − n) Pm+n + km δm+n,0 , i {Jm, ψn} = (sm − n) ψm+n , (7.21) s−1/2 2s−q s−q− 1 2 1 1 2s+1 X (−1) 2 (q) s−q− 2 s− 2km (s) i {ψm, ψn} = h P + (−1) 2 δm+n,0 + Ξ . s − q + 1 k m,n m+n α (2s)! m+n q=0 2 s

– 18 – The conformal weight of the fermionic generators ψn with respect to Jm is given by ∆ = (q) s + 1. Here hm,n stand for the homogeneous polynomials deﬁned through eq. (7.14), extended to the case of (half-)integers, and

r X Pm+n := Pm+n−i1···−ir Pi1 ···Pir . (7.22) i1,···ir

(s) Here Ξm+n stands for the mode expansion of nonlinear terms that contains derivatives of P, and becomes nontrivial provided s > 3/2. Indeed, according to eqs. (D.9) and (4.13), (1/2) (3/2) in the case of supergravity (s = 1/2), and for s = 3/2, one finds that Ξm+n = Ξm+n = 0; 2 while for s = 5/2 it is proportional to the mode expansion of (P0) (see eq. (D.20)). The explicit form of Ξ(7/2) is given in eq. (D.30). As in section5, the asymptotic symmetry algebra (7.21) also implies the existence of nonlinear bounds for the energy. Indeed, making the same assumptions, for m = −n = p, the (fermionic) anticommutator is manifestly positive-definite. Furthermore, since in the “rest frame” the bosonic global charges just correspond to P0, the nonlinear terms described (s) by Ξm+n in the fermionic anticommutator do not contribute. Therefore, in the generic case the bounds are given by n Y 2 πP p2 + (2i + 1) ≥ 0 , (7.23) k i=0 where p is a (half-)integer for the case of fermionic fields of spin s = n + 3/2 that fulfill (anti)periodic boundary conditions. It is then clear that the bounds are fulfilled for configurations with P > 0, as it is the case of cosmological spacetimes. The fact that they never saturate agrees with the nonexistence of globally-defined Killing tensor-spinors. Note that in the case of P = 0 the bounds are also satisfied, while the one with p = 0 saturates, which corresponds to the fact that configurations of this sort admit a single unbroken hypersymmetry, being generated by a constant Killing tensor-spinor. For the class of maximally hypersymmetric smooth solutions with negative energy 2 P = −kj /π described in section 3.1, the bounds (7.23) read

n Y 2 p2 − (2i + 1) j2 ≥ 0 , (7.24) i=0 which implies that the only case that fulfills all of them, also saturate the ones for p = ±(2i + 1)/2, with i = 0, 1, . . . , n, and corresponds to j2 = 1/4. Thus, Minkowski spacetime becomes naturally selected at the ground state in the case of fermions that satisfy antiperiodic boundary conditions, possessing the maximum number of Killing tensor-spinors described by (7.8), with (3.12) and (3.13). In sum, in the case of fermions that fulfill periodic boundary conditions the energy spectrum is nonnegative (P ≥ 0), so that the allowed class of solutions is generically characterized by the cosmological spacetimes described in section 3.1. The ground state is then given by a configuration of vanishing energy that saturates only one of the bounds (p = 0). This corresponds to the null orbifold which, as shown in section 7.1, possesses a

– 19 – single Killing tensor-spinor. If the fermions satisfy antiperiodic boundary conditions, the spectrum becomes enlarged since the bounds now imply that P ≥ −k/4π. Nonetheless, since conical defects and surpluses generically do not fulfill the field equation in vacuum, they are discarded unless they are smooth. According to (7.24), in this case the ground state saturates as many bounds as the maximum number of Killing tensor-spinors, and it can be identified with Minkowski spacetime, so that the spectrum acquires a gap.

8 Final remarks

In the case of fermionic fields of spin s = n + 3/2, the locally hypersymmetric extension of the action Iµ,γ in (6.4), that includes parity-odd terms, can also be formulated as a Chern-Simons theory for the hyper-Poincaré group in (C.2). In order to carry out this task, the invariant bilinear form has to be suitably modified, so that it acquires additional components being determined by eq. (6.8). The gauge field reads

a a α a1...an A = e Pa +ω ˆ Ja + ψa1...an Qα , (8.1) where, as in section6, ωˆa = ωa + γea. Therefore, up to a boundary term, the action of the extended hypergravity theory reduces to k 1 ¯ b a1...an Iµ,γ,ψ = Iµ,γ + iψa ...a D + n + γe Γb ψ , (8.2) n 4π ˆ 1 n 2 being by construction locally invariant under 1 δea = n + i¯ Γaψa1...an , 2 a1...an 1 a a a1...an δω = − n + iγ¯a ...a Γ ψ , (8.3) 2 1 n 1 δψa1...an = D + n + γebΓ a1...an − γe Γ(a1 a2...an)b . 2 b b

Note that the extended hypergravity action (8.2), and its corresponding local hypersymmetry transformations (8.3), agree with the corresponding ones for the locally hypersymmetric extension of General Relativity, given by (7.1) and (7.6), respectively, in the case of µ = γ = 0. Consequently, a suitable set of asymptotically flat boundary conditions for the extended theory is also proposed to be described by gauge fields of the form (4.3), (7.15), and (7.18). The canonical generators of the asymptotic symmetries then reduce to the ones in eq. (6.11), with J˜=J +µP, so that their algebra is readily found to be described by (7.21), but with an additional central extension along the Virasoro subalgebra, precisely as in eq. (6.12). It is worth pointing out that prescribing the asymptotic behaviour of gauge fields to be described by deviations with respect to a reference background that go along the highest weight generators of the algebra, turns out to be a very successful strategy. Indeed, this is not only the case of General Relativity in three spacetime dimensions [28], but it is also so for its locally supersymmetric extension with or without cosmological constant [44], [21],

– 20 – or even when the theory is nonminimally coupled to higher spin fields [45], [46], [47], [26], [27], [19], [48], [20], [17], [18], [49]. One of the interesting features of dealing with hypersymmetry, is that nonlinear bounds for the energy have been shown to naturally emerge from the anticommutator of fermionic generators. In the case of vanishing cosmological constant, the hypersymmetry bounds for the theory in vacuum turn out to exclude solutions that describe conical defects and surpluses [10], [11], despite the latter are maximally (hyper)symmetric. In presence of higher spin fields, the analogue of this class of configurations has been discussed in [50], [51], [7], [52], [53], [54], [55], [9]. It is then worth highlighting that, according to the results that have been recently obtained for hypergravity with negative cosmological constant [9], one is naturally led to expect that only a suitable subset of asymptotically flat solitonic- like solutions might fulfill the hypersymmetry bounds, for which the higher spin charges become tuned in terms of the mass. Indeed, it is amusing to verify that the gauge group Sp (4) ⊗ OSp (1|4) admits an inequivalent Inönü-Wigner contraction as compared with the one described in section 4.1, so that the electric-like spin-4 charges cannot be consistently decoupled in this alternative flat limit of the (0, 1) theory. This contraction is defined through a different change of basis:2 1 Pˆ = Lˆ+ + Lˆ− , Jˆ = Lˆ+ − Lˆ− , i ` i −i i i −i r 1 + − + − 6 + Wˆ n = Uˆ + Uˆ , Vˆn = Uˆ − Uˆ , Qˆp = Sˆ , (8.4) ` n −n n −n ` p ˆ− ˆ− ˆ+ ˆ+ ˆ+ where Li , Un stand for the generators of the (left) sp (4) algebra, and Li , Un , Sp span the (right) osp (1|4) algebra. Thus, in the limit of large AdS radius, ` → ∞, the nonvanishing components of the (anti)commutators of the new algebra read h i h i Jî, Jˆj = (i − j) Jî+j , Jî, Pˆj = (i − j) Pî+j , h i h i Jî, Wˆ n = (3i − n) Wˆ i+n , Jî, Vˆn = (3i − n) Vî+n , h i h i 3i Pˆ , Vˆ = (3i − n) Wˆ , Jˆ , Qˆ = − p Qˆ , i n i+n i p 2 i+p h i 1 2 2 Vˆ , Vˆ = (m − n) m2 + n2 − 4 (m2 + n2 − mn − 9) − (mn − 6) mn Jˆ m n 223 3 3 m+n 1 2 2 + (m − n) m − mn + n − 7 Vˆm+n , (8.5) 6 h i 1 2 2 Vˆ , Wˆ = (m − n) m2 + n2 − 4 (m2 + n2 − mn − 9) − (mn − 6) mn Pˆ m n 223 3 3 m+n 1 + (m − n) m2 − mn + n2 − 7 Wˆ , 6 m+n h i 1 Vˆ , Qˆ = 2m3 − 8m2p + 20mp2 + 82p − 23m − 40p3 Qˆ , m p 233 m+p

2 Note that OSp (1|4), corresponding to the super-AdS4 group, as well as the superconformal group in three spacetime dimensions, admits two interesting consistent “ﬂat limits” (` → ∞), which can be obtained rescaling the generators either as in eq. (4.21), or in (8.4), provided the left copy is switched oﬀ.

– 21 – n o 1 Qˆ , Qˆ = 3Wˆ + 6p2 − 8pq + 6q2 − 9 Pˆ , p q p+q 22 p+q which is to be regarded to span the gauge group of flat hypergravity coupled to spin-4 fields. 1 3 Here, i, j = 0, ±1, m, n = 0, ±1, ±2, ±3, and p, q = ± 2 , ± 2 . The mode expansion of the asymptotically flat symmetry algebra of hypergravity with a fermionic spin-5/2 field, being coupled to spin-4 fields, is then expected to be such that the nonvanishing Poisson brackets are given by 3 i {Jm, Jn} = (m − n) Jm+n , i {Jm, Pn} = (m − n) Pm+n + km δm+n,0 ,

i {Jm, Wn} = (3m − n) Wm+n , i {Jm, Vn} = (3m − n) Vm+n , 3m i {P , V } = (3m − n) W , i {J , ψ } = − n ψ , m n m+n m n 2 m+n 1 i {V , V } = (m − n) 3m4 − 2m3n + 4m2n2 − 2mn3 + 3n4 J m n 2232 m+n 1 233π + (m − n) m2 − mn + n2 V − (m − n)Θ(6) 6 m+n k m+n 72π − (m − n) m2 + 4mn + n2 Θ(4) 32k m+n 1 4 3 2 2 3 4 i {Vm, Wn} = (m − n) 3m − 2m n + 4m n − 2mn + 3n Pm+n (8.6) 2232 1 233π + (m − n) m2 − mn + n2 W − (m − n)Ω(6) 6 m+n k m+n 72π k − (m − n) m2 + 4mn + n2 Ω(4) + m7δ , 32k m+n 2232 m+n,0 1 23π i {V , ψ } = m3 − 4m2n + 10mn2 − 20n3 ψ − iΩ(11/2) m n 223 m+n 3k m+n π + (23m − 82n)Ω(9/2) , 3k m+n 3 4 9π i {ψ , ψ } = 3W + m2 − mn + n2 P + Ω(4) + km4δ , m n m+n 2 3 m+n k m+n m+n,0

(l) (l) 3 where Ωm , and Θm stand for the mode expansion of the following nonlinear terms: 1 Ω(4) = P2 , 2 Θ(4) = JP , 1 Ω(9/2) = Pψ , 2 27 Ω(11/2) = P0ψ , (8.7) 46 7 2π 295 11 Ω(6) = − WP − P3 + (P0)2 + PP00 , 36 3k 864 27 7 2π 295 11 25 Θ(6) = − (VP + WJ ) − P2J + J 0P0 + (JP00 + PJ 00) + iψψ0 . 36 k 432 27 72 3The inﬁnite-dimensional nonlinear algebras in eqs. (4.22) and (8.6), correspond to diﬀerent hypersymmetric extensions of the BMS3 algebra, being isomorphic to the Galilean conformal algebra in two dimensions, and then relevant in the context of non-relativistic holography [56], [57], [16], [58], [59].

– 22 – Indeed, this asymptotic symmetry algebra is recovered from a contraction that corresponds to a different flat limit of W(2,4)⊕W 5 , as compared with the one in 4.1. The flat limit (2, 2 ,4) is now defined according to 1 P = L+ + L− , J = L+ − L− , n ` n −n n n −n r 1 + − + − 6 + Wn = U + U , Vn = U − U , ψn = Ψ , (8.8) ` n −n n −n ` n where the level also rescales as κ = k`. Moreover, once the modes Pm are shifted according to (4.14), it is simple to verify that the wedge algebra of (8.6) reduces to the algebra of the gauge group in (8.5). From the anticommutator of the fermionic generators in (8.6), one then finds that the zero modes of the energy and the electric-like spin-4 charge, 2πP = P0, 2πW = W0, fulfill the following bounds 9π k 3W + P2 + 5p2P + p4 ≥ 0 , (8.9) 2k 2π which agree with the bounds in [9] in the flat limit. It would then be interesting to explore different classes of solutions endowed with electric-like spin-4 charge, including cosmological spacetimes and solitonic-like configurations that fulfill the bounds (8.9), as well as the hypersymmetric ones that should saturate them. Note that since the bounds (8.9) factorize as 2 2 2 2 p + λ[+] p + λ[−] ≥ 0 , (8.10) it is natural to expect that the eigenvalues of the holonomy of the dynamical gauge field aφ along an angular cycle, for the class of solutions aforementioned, have to be given by r ! 5π 4 3k λ2 = P ± P2 − W . (8.11) [±] k 5 8π

In the case of solitonic-like solutions, these eigenvalues should then correspond to a couple of purely imaginary integers, that become related for the class of configurations that fulfill the bounds (8.10), saturating just some of them. As a final remark, since the hyper-Poincaré group actually exists for d ≥ 3 spacetime dimensions [6], it would be interesting to explore whether similar results as the ones obtained here could extend to higher-dimensional spacetimes. In this sense, despite the no-go results in four dimensions [2], [3], [60], [61], [62], some interesting results have been recently found in the case of hypergravity at the noninteracting level [63]. Whether these results correspond to a suitable weak field limit of Vasiliev higher spin gravity [64], [65], or another theory that has yet to be found, remains as an open question.

Acknowledgments

We thank M. Henneaux, A. Pérez and D. Tempo for helpful discussions. We also thank the International Solvay Institutes and the ULB for warm hospitality. O.F. and R.T. wish to

– 23 – thank Amilcar de Queiroz, Emanuele Orazi, and the organizers of the school and conference “Theoretical Frontiers in Black Holes and Cosmology”, hosted by IIP-UFRN, during June 2015, for the opportunity of presenting this work. O.F. thanks Conicyt for ﬁnancial sup- port. This research has been partially supported by Fondecyt grants Nº 1130658, 1121031, 3150448. The Centro de Estudios Cientíﬁcos (CECs) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt.

A Conventions

We follow the conventions of [6]. The orientation is chosen to be such that the Levi-Civita symbol fulﬁlls ε012 = 1, and the Minkowski metric is assumed to be non-diagonal, whose only nonvanishing components read η01 = η10 = η22 = 1. Round brackets correspond to symmetrization of the indices enclosed by them, so that

X(a|Y bZ|c) = XaY bZc + XcY bZa . (A.1)

Note that the three-dimensional Dirac matrices, that satisfy the Cliﬀord algebra {Γa, Γb} = 2ηab, fulﬁll the following identity:

ΓaΓbΓc = εabc + ηabΓc + ηbcΓa − ηacΓb . (A.2)

The generators of the Lorentz group, in the spinorial and vector (adjoint) representations, α 1 α b b are assumed to be given by (Ja)β = 2 (Γa)β , and (Ja)c = −εa c, respectively. The three-dimensional Γ-matrices are chosen as 1 1 Γ0 = √ (σ1 + iσ2) , Γ1 = √ (σ1 − iσ2) , Γ2 = σ3 , (A.3) 2 2 where σi stand for the Pauli matrices: ! ! ! 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . (A.4) 1 0 i 0 0 −1

α For a vector-spinor ψa , with α = +, −, and a = 0, 1, 2, the Majorana conjugate is deﬁned ¯ β as ψαa = ψa Cβα, where the charge conjugation matrix C, and its inverse are chosen as ! ! 0 −1 αβ 0 1 Cαβ = ,C = , (A.5) 1 0 −1 0

T T so that C = −C, and (CΓa) = CΓa.

B Killing vector-spinors from an alternative approach

α Killing vector-spinors a in (3.2) that are globally well deﬁned can be recovered as a particular case of the asymptotic symmetries discussed in section4. Indeed, they are of the α a form a Qα = λ [0, 0, E], with λ given by (4.5). Hence, for the class of bosonic conﬁgurations

– 24 – discussed in section 3.1, that only carries the zero modes of the global charges, and their corresponding chemical potentials are constant, the components of the Killing vector-spinors can be written as ˆ 0 ˆ 1 3π 00 ˆ π 7 0 k 000 ˆ λ [0, 0, E] = EQ 3 − E Q 1 − EP − E Q− 1 − − E P + E Q− 3 . (B.1) 2 2 2 k 2 3k 2 2π 2

The requirements of invariance under hypersymmetry can then be read from eqs. (4.7), (4.10), so that the Killing vector-spinor equation reduces to

9π k δψ = − P2E + 5PE00 − E0000 = 0 , (B.2) 2k 2π

˙ 0 E = µJ E , (B.3) which can be readily integrated. In fact, the solution of eq. (B.2) is generically given by

q q q q πP φ − πP φ 3 πP φ −3 πP φ E = A1e k + A2e k + A3e k + A4e k , (B.4) where AI = AI (u) stand for four arbitrary functions. In the case of P > 0, E clearly cannot fulﬁll neither antiperiodic nor periodic boundary conditions, and therefore, cosmological spacetimes break all the hypersymmetries. Note that if the energy vanishes (P = 0), eq. (B.2) integrates in a diﬀerent way:

2 3 E = A0 + A1φ + A2φ + A3φ . (B.5)

Periodicity then implies that E = A0 (u), while the remaining equation (B.3) ﬁxes the arbitrary function to be a constant. Hence, vanishing energy conﬁgurations, as the null orbifold, admit a single constant Killing vector-spinor. Finally, for P = −kn2/π < 0, eq. (B.4) reads

inφ ∗ −inφ 3inφ ∗ −3inφ E = A1e + A1e + A3e + A3e , (B.6) so that n turns out to be a (half-)integer for fermions fulﬁlling (anti)periodic boundary conditions. The remaining equation (B.3) then ﬁxes the form of the arbitrary functions AI (u), and hence there are four independent Killing vector-spinors, determined by

in(µJ u+φ) ∗ −in(µJ u+φ) 3in(µJ u+φ) ∗ −3in(µJ u+φ) E = E1e + E1 e + E3e + E3 e . (B.7)

It is worth pointing out that Minkowski spacetime, which corresponds to n2 = 1/4, is the only one of this class that fulﬁlls all the hypersymmetry bounds (5.3), saturating precisely four of them in the case of antiperiodic boundary conditions. Indeed, the remaining solutions of this sort, in spite of possessing the maximum number of hypersymmetries, become manifestly excluded by the bounds. In the case of periodic boundary conditions, the null orbifold also satisﬁes the bounds, but it saturates only one of them.

– 25 – 1 C Hyper-Poincaré algebra with fermionic generators of spin n + 2

The extension of the Poincaré algebra in the generic case includes fermionic tensor-spinor a1...an generators Qα , being completely symmetric in the vector indices, and Γ-traceless, i. e., a1...an Q Γa1 = 0 [6]. The Maurer-Cartan 1-form,

a a α a1...an Ω = ρ Pa + τ Ja + χa1...an Qα , (C.1) corresponds to a ﬂat connection that fulﬁlls

a 1 abc dτ = − τbτc , (C.2) 2 1 1 dρa = −abcτ ρ + n + iχ¯ Γaχa1...an , b c 2 2 a1...an 1 dχa1...an = − n + τ bΓ χa1...an + τ Γ(a1|χ|a2...an)b , 2 b b where χa1...an is Γ-traceless, and completely symmetric in the vector indices. Apart from a I1 = P Pa, this algebra admits an additional Casimir operator given by

a αβ a1...an I2 = 2J Pa + Qαa1...an C Qβ . (C.3)

D Asymptotic hypersymmetry algebra

D.1 Spin-3/2 ﬁelds (supergravity) In our conventions, the super-Poincaré algebra with N = 1 reads

c [Ja,Jb] = εabcJ , c [Ja,Pb] = εabcP , (D.1) 1 β [Ja,Qα] = (Γa) Qβb , (D.2) 2 α 1 {Q ,Q } = − (CΓc) P , α β 2 αβ c so that changing the basis according to

Jˆ−1 = −2J0 , Jˆ1 = J1 , Jˆ0 = J2 ,

Pˆ−1 = −2P0 , Pˆ1 = P1 , Pˆ0 = P2 , (D.3) 3 1 Qˆ 1 = 2 4 Q+ , Qˆ 1 = 2 4 Q− , − 2 2 it acquires the following form h i Jˆm, Jˆn = (m − n) Jˆm+n , h i Jˆm, Pˆn = (m − n) Pˆm+n , h i m Jˆm, Qˆp = − p Qˆm+p , (D.4) 2 n o Qˆp, Qˆq = −Pˆp+q .

– 26 – 1 with m, n = 0, ±1, and p, q = ± 2 . The asymptotic form of the dynamical gauge ﬁeld then reads ˆ π ˆ ˆ ˆ aφ = J1 − PJ−1 + J P−1 + ψQ− 1 , (D.5) k 2 which is mapped into itself under gauge transformations generated by ˆ ˆ ˆ λ = T P1 + Y J1 + EQ 1 + η 1 [T,Y, E] , (D.6) 2 ( 2 ) with 0 ˆ 0 ˆ 0 πY ψ ˆ 1 00 2πY P ˆ η 1 [T,Y, E] = −T P0 − Y J0 − E + Q− 1 + Y − J−1 ( 2 ) k 2 2 k 1 00 2πT P 2πY J iπEψ + T − − + Pˆ−1 , (D.7) 2 k k k provided the ﬁelds transform according to: k δP = 2PY 0 + P0Y − Y 000 , 2π k 3 1 δJ = 2J Y 0 + J 0Y + 2PT 0 + P0T − T 000 + iψE0 + iψ0E , (D.8) 2π 2 2 3 k δψ = ψY 0 + ψ0Y − PE + E00 . 2 π Once expanded in modes, the asymptotic symmetry algebra is found to be given by

i {Jm, Jn} = (m − n) Jm+n , 3 i {Jm, Pn} = (m − n) Pm+n + km δm+n,0 , m i {Jm, ψn} = − n ψm+n , (D.9) 2 2 i {ψm, ψn} = Pm+n + 2km δm+n,0 , and hence in the case of fermions that fulﬁll periodic boundary conditions, the energy P0 P = 2π is bounded to be nonnegative, P ≥ 0 , (D.10) while in the case of fermions subject to antiperiodic boundary conditions, the energy fulﬁlls 1 πP + ≥ 0 . (D.11) 4 k These results agree with the ones in [21].

D.2 Spin-7/2 ﬁelds In the case of fermionic generators of (conformal) spin ∆ = 7/2, the hyper-Poincaré algebra can be written as h i Jˆm, Jˆn = (m − n) Jˆm+n , h i Jˆm, Pˆn = (m − n) Pˆm+n , (D.12) h i 5m Jˆm, Qˆp = − p Qˆm+p , (D.13) 2 n ˆ ˆ o (5/2) ˆ Qp, Qq = fp,q Pp+q ,

– 27 – with

1 f (5/2) = − 80 p4 + q4 − 128 p3q + pq3 p,q 192 2 2 2 2 +144p q − 620 p + q + 832pq + 675 , (D.14)

1 3 5 where m, n = 0, ±1, and p, q = ± 2 , ± 2 , ± 2 . The asymptotic behaviour of the dynamical gauge ﬁeld now reads

ˆ π ˆ ˆ ψ ˆ aφ = J1 − J P−1 + PJ−1 + Q− 5 , (D.15) k 10 2 so that the asymptotic symmetries are now spanned by

ˆ ˆ ˆ λ = T P1 + Y J1 + EQ 1 + η 5 [T,Y, E] , (D.16) 2 ( 2 ) with

0 ˆ 0 ˆ 0 ˆ 1 00 2πT P 2πY J 5iπEψ ˆ η 5 [T,Y, E] = −T P0 − Y J0 − E Q 3 + T − − − P−1 ( 2 ) 2 2 k k k 1 00 2πY P 1 00 5πEP + Y − Jˆ−1 + E − Qˆ 1 2 k 2 k 2 0 0 1 000 13πE P 5πEP ˆ − E − − Q− 1 (D.17) 6 k k 2 2 2 00 0 0 00 1 (4) 45π EP 22πE P 18πE P 5πEP ˆ + E + − − − Q− 3 24 k2 k k k 2 1 149π2E0P2 12πY ψ 30πE000P − E(5) + + − 120 k2 k k 2 0 00 0 0 00 000 130π EPP 40πE P 23πE P 5πEP ˆ + − − − Q− 5 , k2 k k k 2 and the transformation law of the ﬁelds is given by

k δP = 2PY 0 + P0Y − Y 000 , 2π k 7 5 δJ = 2J Y 0 + J 0Y + 2PT 0 + P0T − T 000 + iψE0 + iψ0E , (D.18) 2π 2 2 ! 7 75π2P3 65π (P0)2 155πPP00 5 δψ = ψY 0 + ψ0Y + − + + − P(4) E 2 4k2 6k 12k 12 259πPP0 7 259πP2 21 35 35 kE(6) + − P000 E0 + − P00 E00 − E000P0 − E(4)P + . 6k 3 12k 4 6 12 12π

– 28 – The Poisson brackets of the asymptotic symmetry algebra in this case read

i {Jm, Jn} = (m − n) Jm+n , 3 i {Jm, Pn} = (m − n) Pm+n + km δm+n,0 , 5m i {Jm, ψn} = − n ψm+n , (D.19) 2 1 i {ψ , ψ } = 5m4 − 8m3n + 9m2n2 − 8mn3 + 5n4 P m n 12 m+n 1 X + 155m2 − 208mn + 155n2 P P 48k m+n−q q q 75 X k (5/2) + P P P + m6δ + Ξ , 16k2 m+n−q−r q r 6 m+n,0 m+n q

(5/2) (5/2) −imφ where Ξm = Ξ e dφ stands for the mode expansion of ´ 25π (P0)2 Ξ(5/2) = . (D.20) 12k The anticommutator of the fermionic charges then implies that the energy has to fulﬁll the following bounds 25πP 9πP πP p2 + p2 + p2 + ≥ 0 , (D.21) k k k with p given by a (half-)integer for fermions that fulﬁll (anti)periodic boundary conditions.

D.3 Spin-9/2 ﬁelds

The hyper-Poincaré algebra with fermionic generators of (conformal) spin ∆ = 9/2 is described by h i Jˆm, Jˆn = (m − n) Jˆm+n , h i Jˆm, Pˆn = (m − n) Pˆm+n , (D.22) h i 7m Jˆm, Qˆp = − p Qˆm+p , (D.23) 2 n ˆ ˆ o (7/2) ˆ Qp, Qq = fp,q Pp+q , where 1 f (7/2) = 112 p6 + q6 − 192 p5q + pq5 + 240 p4q2 + p2q4 − 256p3q3 p,q 2304 4 4 3 3 2 2 −2240 p + q + 3616 p q + pq − 4080p q (D.24) +11578 p2 + q2 − 15592pq − 11025 ,

1 3 5 7 and with m, n = 0, ±1, and p, q = ± 2 , ± 2 , ± 2 , ± 2 .

– 29 – The asymptotic fall-oﬀ of the dynamical gauge ﬁeld is now given by

ˆ π ˆ ˆ ψ ˆ aφ = J1 − J P−1 + PJ−1 − Q− 7 , (D.25) k 35 2 and the asymptotic symmetries turn out to be parametrized according to

ˆ ˆ ˆ λ = T P1 + Y J1 + EQ 1 + η 7 [T,Y, E] , (D.26) 2 ( 2 ) with

0 ˆ 0 ˆ 0 ˆ 1 00 2πT P 2πY J 7iπEψ ˆ η 7 [T,Y, E] = −T P0 − Y J0 − E Q 5 + T − − − P−1 ( 2 ) 2 2 k k k 0 0 1 00 2πY P 1 00 7πEP 1 000 19πE P 7πEP + Y − Jˆ−1 + E − Qˆ 3 − E − − Qˆ 1 2 k 2 k 2 6 k k 2 2 2 00 0 0 00 1 (4) 105π EP 34πE P 26πE P 7πEP ˆ + E + − − − Q− 1 24 k2 k k k 2 1 409π2E0P2 322π2EPP0 50πE000P 60πE00P0 − E(5) + + − − 120 k2 k2 k k 0 00 000 3 3 33πE P 7πEP ˆ 1 (6) 1575π EP − − Q− 3 + E − (D.27) k k 2 720 k3 919π2E00P2 1530π2E0PP0 427π2EPP00 322π2E (P0)2 + + + + k2 k2 k2 k2 (4) (3) 0 00 00 0 000 (4) ! 65πE P 110πE P 93πE P 40πE P 7πEP ˆ − − − − − Q− 5 k k k k k 2 1 6483π3E00P3 8589π3EP2P0 1519π2E000P2 − E(7) − − + 5040 k3 k3 k2 4088π2E00PP0 2353π2E0PP00 1852π2E0 (P0)2 + + + k2 k2 k2 511π2EPP000 1071π2EP0P00 144πY ψ 77πE(5)P 175πE(4)P0 + + − − − k2 k2 k k k 000 00 00 000 0 (4) (5) ! 203πE P 133πE P 47πE P 7πEP ˆ − − − − Q− 7 , k k k k 2 so that the ﬁelds transform as

k δP = 2PY 0 + P0Y − Y 000 , 2π k 9 7 δJ = 2J Y 0 + J 0Y + 2PT 0 + P0T − T 000 + iψE0 + iψ0E , (D.28) 2π 2 2

– 30 –

9 1225π3P4 5789π2P00P2 2429π2 (P0)2 P δψ = ψY 0 + ψ0Y + − + + 2 16k3 72k2 18k2 ! 35πP(4)P 119π (P00)2 791πP0P000 7 − − − + P(6) E + 9k 16k 72k 144 3229π2P0P2 131πP000P 99πP0P00 3 + − − + P(5) E0 12k2 6k 2k 8 ! 3229π2P3 197πP00P 165π (P0)2 5 + − − + P(4) E00 36k2 4k 4k 4 7 329πPP0 21 329πP2 + P000 − E(3) + P00 − E(4) 3 6k 8 24k 7 7 kE(8) + E(5)P0 + E(6)P − . 4 12 144π

The mode expansion of the asymptotic symmetry algebra is then given by

i {Jm, Jn} = (m − n) Jm+n , 3 i {Jm, Pn} = (m − n) Pm+n + km δm+n,0 , 7m i {Jm, ψn} = − n ψm+n , (D.29) 2 1 i {ψ , ψ } = 7m6 − 12m5n + 15m4n2 − 16m3n3 + 15m2n4 − 12mn5 + 7n6 P m n 144 m+n 1 X + 140m4 − 226m3n + 255m2n2 − 226mn3 + 140n4 P P 144k m+n−q q q 1 X + 5789m2 − 7796mn + 5789n2 P P P 864k2 m+n−q−r q r q,r 1225 X k (7/2) + P P P P + m8δ + Ξ , 128k3 m+n−q−r q r t 72 m+n,0 m+n q,r,t where (7/2) 2 2 Ξm+n = 7Θm+n + 329m − 494mn + 329n χm+n , (D.30) and Θm and χm correspond to the mode expansion of

1596π2P (P0)2 661π (P00)2 π (P0)2 Θ = + , χ = , (D.31) 432k2 432k 144k respectively. The energy is then found to fulﬁll the following bounds

49πP 25πP 9πP πP p2 + p2 + p2 + p2 + ≥ 0 , (D.32) k k k k where according to the (anti)periodicity conditions for the fermions, p corresponds to a (half-)integer.

– 31 – References

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